Tests for independence of rows and columns in an $r \times s$ contingency table are developed from canonical correlation analysis and from models of linear-by-linear interaction. The resulting test statistics are asymptotically equivalent under the null hypothesis. They are consistent and asymptotically unbiased. Approximate critical values are available from existing tables. The proposed tests are most appropriate when the matrix of joint probabilities is well approximated by a matrix of rank 2. Against some alternatives which may arise in such tables, the proposed statistics have greater asymptotic power than conventional chi-square tests of independence.
"Tests for Independence in Two-Way Contingency Tables Based on Canonical Correlation and on Linear-By-Linear Interaction." Ann. Statist. 9 (6) 1178 - 1186, November, 1981. https://doi.org/10.1214/aos/1176345635